### Current Transformer Magnetizing Curve

Instrument Transformer and Power Management (P1) Course
Chapter (6) : Current Transformers
6.3.4 Current Transformer Magnetizing Curve
The primary current contains two components. These are respectively the Secondary current which is transformed in the inverse ratio of the turns ratio and an exciting current, which supplies the eddy and hysteresis losses and magnetizes the core.
This latter current flow in the primary winding only and therefore, is the cause of the transformer errors. It is, therefore, not sufficient to assume a value of secondary current and to work backwards to determine the value of primary current by invoking the constant ampere -turns rule. Since this approach does not take into account the exciting current. From this observation it may be concluded that certain values of secondary current could never be produced whatever the value of primary current and this is of course, the case when the core saturates and a disproportionate amount of primary current is required to magnetize the core.

The amount of exciting current drawn by a current transformer depends upon the core material and the amount of flux which must be developed in the core to satisfy the burden requirements of the current transformer.

The appropriate current may be obtained directly from the exciting characteristic of the transformer since the secondary E.M.F. and therefore the flux developed is proportional to the product of secondary current and burden impedance.

The general shape of the exciting characteristic for atypical grade of CROSS (cold rolled grain orientated silicon steel) is shown. The characteristic is divided into three regions, defined by the " ankle -point " and the " knee -point ".

The working range of a protective current transformer extends over the full range between the " ankle -point " and the " knee -point " and beyond, while measuring current transformer usually only operates in the region of the ankle-point ".
The current transformer mangnetization curve, is usually expressed in terms of Kv and Ki which when multiplied by the flux density in tesla and ampere-turns per cm respectively gives corresponding volts and amperes.
Es = 4.44 FBAN volts.
In this equation, the flux density B is in teals and the core cross sectional area is in square meters.

Knee-Point :
The knee -point of the excitation characteristic is defined as the point at which a 10% increase in secondary voltage produces a 50% increase in exciting current. It may, therefore be regarded as a practical limit beyond which a specified current ratio may not be maintained.

Beyond the knee point the current transformer is said to enter saturation. In this region the major part of the primary current is utilized to maintain the core flux, and since the shunt admittance is not linear, both the exciting and secondary currents depart from a sine wave.

For example :
In the case of wholly resistive burden, correct transformation takes place until saturation flux density is reached, The secondary volts and current then collapse instantly to zero, where they remain until next primary current zero is reached, This process is repeated each half cycle and results in a pulse waveform as shown.

6.3.5 Open Circuited Secondary Winding :
The secondary circuit of a current transformer should never be left open circuited whilst primary current continues to flow. In these circumstances only the primary winding is effective and thus the current transformer behaves as a highly saturated choke ( induction) to the flow of primary winding current.

Thus a peaky and relatively high value of voltage appears in the secondary output of terminals, endangering life, not to mention the possible resulting breakdown of secondary circuit insulation.

In those cases where current transformers are associated with the " high impedance type " earth fault relay the secondary circuit burden may have ohmic values up to several thousands of ohms.

In these circumstances the maximum crest value of any secondary over voltage which may occur may be computed by the following expression :
VP =  2 x √2  x √((VTH - VK ) x VK)
Where : VTH = The r.m.s voltage required to circulate the maximum available current which would flow in the secondary circuit if core saturation had not occurred.
VK = knee-point voltage of current transformers.

Where the value of VP exceeds 3 KV it is considered prudent to connect a non linear resistor ( METROSIL ) across a secondary output terminals.

From what has been discussed above it will be evident that :
" Where a current transformer is not in use but where its primary winding is still energized it is important to ensure that the secondary winding is left "short circuited "

6.3.6 Equivalent Circuit :
The errors of a current transformer may be considered as due to the whole of the primary current not being transformed , a component therefore being required to excite the core.

Alternatively, we may consider that the whole of the primary current is transformed without loss, but that the secondary current is shunted by a parallel circuit, the impedance of which is such that the equivalent of the exciting current flows there in. The circuit shown in the equivalent circuit of the current transformer. The primary current is assumed to be transformed perfectly, with no ratio or phase angle error, to a current (Ip/N) which is often called the primary current referred to the secondary.

A part of current may be considered consumed in exciting the core, and this current Ie is called the secondary excitation current. The remainder Is is the true secondary current it will be evident that, the excitation current is a function of the secondary excitation voltage Es and the secondary excitation impedance Ze.

It will also be evident that the secondary current is a function of the secondary excitation voltage Es and the total impedance in the secondary circuit. This total impedance consists of the effective resistance ( and any leakage reactance) of the secondary winding and the impedance of the burden.
Ip = Primary current in amperes.
N = Current transformer ratio ( primary to secondary amperes ).
Zb = Burden impedance of relay in ohms ( r + jx ).
Zs = Current transformer secondary winding impedance in ohms (r + jx).
Ze = Secondary excitation impedance in ohms ( jx ).
le = Secondary excitation current in amperes.
Is = Secondary current in amperes.
Es = Secondary excitation voltage in volts.
Vt = Secondary terminal voltage in volts across the current transformer terminals ( input to the relay or burden ).